Abstract
We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamiltonians with some partial coercivity and some related restrictions on the oscillating variables, mostly motivated by the applications to differential games, in particular of pursuit-evasion type. The effective initial data are computed under some assumptions of asymptotic controllability of the underlying control system with two competing players.
Highlights
We are concerned with the homogenization of Hamilton-Jacobi equations with oscillating initial data ∂tuε + H (z, z ε Dz uε ) =in (0, ∞) × RN uε(0, z) h(z, z ε ) in RN (1)with Hamiltonian H and initial data h at least continuous and ZN -periodic in the second entry where the oscillating variables appear
In this paper we focus our attention on the Hamiltonians of Bellman-Isaacs type that arise in deterministic optimal control and in the theory of two-person, zero-sum differential games, namely
H1 and H2 are coercive we prove homogenization for h that either has a saddle or is of the form h h1(x, y, x−y ε h2(x, y, y ε
Summary
With Hamiltonian H and initial data h at least continuous and ZN -periodic in the second entry where the oscillating variables appear. The goal is finding an effective Hamiltonian H : RN × RN → R and effective initial data h : RN → R such that uε converges (locally uniformly) as ε → 0 to the solution of. ∂tuε + H (z, Dzuε) = 0 in (0, ∞) × RN uε(0, z) = h(z) in RN. Homogenization, Hamilton-Jacobi equations, Isaacs equations, viscosity solutions, differential games, oscillating initial data.
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